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Properties

Label 2-140-28.3-c1-0-3
Degree $2$
Conductor $140$
Sign $0.822 - 0.569i$
Analytic cond. $1.11790$
Root an. cond. $1.05731$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.501 − 1.32i)2-s + (−0.895 + 1.55i)3-s + (−1.49 + 1.32i)4-s + (0.866 − 0.5i)5-s + (2.49 + 0.405i)6-s + (−0.644 + 2.56i)7-s + (2.50 + 1.31i)8-s + (−0.103 − 0.179i)9-s + (−1.09 − 0.894i)10-s + (3.66 + 2.11i)11-s + (−0.717 − 3.50i)12-s + 2.98i·13-s + (3.71 − 0.434i)14-s + 1.79i·15-s + (0.480 − 3.97i)16-s + (−1.92 − 1.10i)17-s + ⋯
L(s)  = 1  + (−0.354 − 0.934i)2-s + (−0.516 + 0.895i)3-s + (−0.748 + 0.663i)4-s + (0.387 − 0.223i)5-s + (1.02 + 0.165i)6-s + (−0.243 + 0.969i)7-s + (0.885 + 0.464i)8-s + (−0.0344 − 0.0596i)9-s + (−0.346 − 0.282i)10-s + (1.10 + 0.638i)11-s + (−0.207 − 1.01i)12-s + 0.827i·13-s + (0.993 − 0.116i)14-s + 0.462i·15-s + (0.120 − 0.992i)16-s + (−0.465 − 0.268i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.822 - 0.569i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.822 - 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.822 - 0.569i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 140,\ (\ :1/2),\ 0.822 - 0.569i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.735656 + 0.229902i\)
\(L(\frac12)\) \(\approx\) \(0.735656 + 0.229902i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.501 + 1.32i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (0.644 - 2.56i)T \)
good3 \( 1 + (0.895 - 1.55i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (-3.66 - 2.11i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.98iT - 13T^{2} \)
17 \( 1 + (1.92 + 1.10i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.28 + 3.95i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.78 - 1.02i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.42T + 29T^{2} \)
31 \( 1 + (-1.20 + 2.07i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.16 - 3.74i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.88iT - 41T^{2} \)
43 \( 1 + 12.3iT - 43T^{2} \)
47 \( 1 + (-3.38 - 5.85i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.41 + 11.1i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.99 + 12.1i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.0195 + 0.0113i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.38 - 2.53i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.07iT - 71T^{2} \)
73 \( 1 + (2.88 + 1.66i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.14 - 1.81i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 11.7T + 83T^{2} \)
89 \( 1 + (14.4 - 8.34i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 12.0iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.97790636574160106440532855589, −11.89950495971581256063659278781, −11.33340342927075345492910495889, −10.04931076746415016214874678667, −9.414462964267419734421238061711, −8.603960552101396376822303197169, −6.66267458011703046201351975586, −5.05116053571708070744855867607, −4.12654000811707338000284151119, −2.19581218299141792295966234456, 1.05125917738979327417903620168, 4.05452377299971480757075151577, 5.93235815690483465018803923134, 6.51338532341866743644675805845, 7.47224273632261544926297982805, 8.636965013076370411934651814441, 9.958811122531885344285258774378, 10.81872373511511139228421692468, 12.27199381812773375954270536427, 13.27486830845068431403431458109

Graph of the $Z$-function along the critical line